Spanning 2-Forests and Resistance Distance in 2-Connected Graphs

A spanning 2-forest separating vertices $u$ and $v$ of an undirected connected graph is a spanning forest with 2 components such that $u$ and $v$ are in distinct components. Aside from their combinatorial significance, spanning 2-forests have an important application to the calculation of resistance distance or effective resistance. The resistance distance between vertices $u$ and $v$ in a graph representing an electrical circuit with unit resistance on each edge is the number of spanning 2-forests separating $u$ and $v$ divided by the number of spanning trees in the graph. There are also well-known matrix theoretic methods for calculating resistance distance, but the way in which the structure of the underlying graph determines resistance distance via these methods is not well understood. For any connected graph $G$ with a 2-separator separating vertices $u$ and $v$, we show that the number of spanning trees and spanning 2-forests separating $u$ and $v$ can be expressed in terms of these same quantities for the smaller separated graphs, which makes computation significantly more tractable. An important special case is the preservation of the number of spanning 2-forests if $u$ and $v$ are in the same smaller graph. In this paper we demonstrate that this method of calculating resistance distance is more suitable for certain structured families of graphs than the more standard methods. We apply our results to count the number of spanning 2-forests and calculate the resistance distance in a family of Sierpinski triangles and in the family of linear 2-trees with a single bend.